Introduction to Tensor Calculus for General Relativity - Part I
Introduction to Tensor Calculus for General Relativity - Part I
Introduction to Tensor Calculus for General Relativity - Part I
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1<br />
Massachusetts Institute of Technology<br />
Department o f P h ysics<br />
Physics 8.962 Spring 2000<br />
<strong>Introduction</strong> <strong>to</strong> <strong>Tensor</strong> <strong>Calculus</strong> <strong>for</strong><br />
<strong>General</strong> <strong>Relativity</strong><br />
c2000 Edmund Bertschinger.<br />
1<br />
<strong>Introduction</strong><br />
There are three essential ideas underlying general relativity (GR). The rst is that spacetime<br />
may be described as a curved, four-dimensional mathematical structure called a<br />
pseudo-Riemannian manifold. In brief, time and space <strong>to</strong>gether comprise a curved fourdimensional<br />
non-Euclidean geometry. Consequently, the practitioner of GR must be<br />
familiar with the fundamental geometrical properties of curved spacetime. In particular,<br />
the laws of physics must be expressed in a <strong>for</strong>m that is valid independently of any<br />
coordinate system used <strong>to</strong> label points in spacetime.<br />
The second essential idea underlying GR is that at every spacetime point there exist<br />
locally inertial reference frames, corresponding <strong>to</strong> locally at coordinates carried by freely<br />
falling observers, in which t h e p h ysics of GR is locally indistinguishable from that of<br />
special relativity. This is Einstein's famous strong equivalence principle and it makes<br />
general relativity an extension of special relativity t o a c u r v ed spacetime. The third key<br />
idea is that mass (as well as mass and momentum ux) curves spacetime in a manner<br />
described by the tensor eld equations of Einstein.<br />
These three ideas are exemplied by c o n trasting GR with New<strong>to</strong>nian gravity. I n t h e<br />
New<strong>to</strong>nian view, gravity is a <strong>for</strong>ce accelerating particles through Euclidean space, while<br />
time is absolute. From the viewpoint of GR as a theory of curved spacetime, there is no<br />
gravitational <strong>for</strong>ce. Rather, in the absence of electromagnetic and other <strong>for</strong>ces, particles<br />
follow the straightest possible paths (geodesics) through a spacetime curved by mass.<br />
Freely falling particles dene locally inertial reference frames. Time and space are not<br />
absolute but are combined in<strong>to</strong> the four-dimensional manifold called spacetime.<br />
In special relativity there exist global inertial frames. This is no longer true in the<br />
presence of gravity. H o wever, there are local inertial frames in GR, such that within a
2<br />
suitably small spacetime volume around an event (just how small is discussed e.g. in<br />
MTW Chapter 1), one may c hoose coordinates corresponding <strong>to</strong> a nearly-at spacetime.<br />
Thus, the local properties of special relativity c a r r y o ver <strong>to</strong> GR. The mathematics of<br />
vec<strong>to</strong>rs and tensors applies in GR much as it does in SR, with the restriction that vec<strong>to</strong>rs<br />
and tensors are dened independently at each spacetime event (or within a suciently<br />
small neighborhood so that the spacetime is sensibly at).<br />
Working with GR, particularly with the Einstein eld equations, requires some understanding<br />
of dierential geometry. In these notes we w i l l d e v elop the essential mathematics<br />
needed <strong>to</strong> describe physics in curved spacetime. Many p h ysicists receive their<br />
introduction <strong>to</strong> this mathematics in the excellent boo k o f W einberg (1972). Weinbe r g<br />
minimizes the geometrical content of the equations by representing tensors using component<br />
notation. We believe that it is equally easy <strong>to</strong> work with a more geometrical<br />
description, with the additional benet that geometrical notation makes it easier <strong>to</strong> distinguish<br />
physical results that are true in any coordinate system (e.g., those expressible<br />
using vec<strong>to</strong>rs) from those that are dependent on the coordinates. Because the geometry<br />
of spacetime is so intimately related <strong>to</strong> physics, we believe that it is better <strong>to</strong> highlight<br />
the geometry from the outset. In fact, using a geometrical approach a l l o ws us <strong>to</strong> develop<br />
the essential dierential geometry as an extension of vec<strong>to</strong>r calculus. Our treatment<br />
is closer <strong>to</strong> that Wald (1984) and closer still <strong>to</strong> Misner, Thorne and Wheeler (1973,<br />
MTW). These books are rather advanced. For the newcomer <strong>to</strong> general relativity w e<br />
warmly recommend Schutz (1985). Our notation and presentation is patterned largely<br />
after Schutz. It expands on MTW Chapters 2, 3, and 8. The student wishing additional<br />
practice problems in GR should consult Lightman et al. (1975). A slightly more<br />
advanced mathematical treatment is provided in the excellent notes of Carroll (1997).<br />
These notes assume familiarity with special relativity. W e will adopt units in which<br />
the speed of light c = 1 . Greek indices (, , etc., which t a k e the range f0 1 2 3g)<br />
will be used <strong>to</strong> represent components of tensors. The Einstein summation convention<br />
is assumed: repeated upper and lower indices are <strong>to</strong> be summed over their ranges,<br />
e.g., A B A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 . Four-vec<strong>to</strong>rs will be represented with<br />
an arrow o ver the symbol, e.g., A, ~ while one-<strong>for</strong>ms will be represented using a tilde,<br />
e.g., B. ~ Spacetime points will be denoted in boldface type e.g., x refers <strong>to</strong> a point<br />
with coordinates x . Our metric has signature +2 the at spacetime Minkowski metric<br />
components are = diag(;1 +1 +1 +1).<br />
2<br />
Vec<strong>to</strong>rs and one-<strong>for</strong>ms<br />
The essential mathematics of general relativity is dierential geometry, the branch o f<br />
mathematics dealing with smoothly curved surfaces (dierentiable manifolds). The<br />
physicist does not need <strong>to</strong> master all of the subtleties of dierential geometry in order
3<br />
<strong>to</strong> use general relativity. ( F or those readers who want a deeper exposure <strong>to</strong> dierential<br />
geometry, see the introduc<strong>to</strong>ry texts of Lovelock and Rund 1975, Bishop and Goldberg<br />
1980, or Schutz 1980.) It is sucient <strong>to</strong> develop the needed dierential geometry as a<br />
straight<strong>for</strong>ward extension of linear algebra and vec<strong>to</strong>r calculus. However, it is important<br />
<strong>to</strong> keep in mind the geometrical interpretation of physical quantities. For this reason,<br />
we will not shy from using abstract concepts like p o i n ts, curves and vec<strong>to</strong>rs, and we will<br />
distinguish between a vec<strong>to</strong>r A ~ and its components A . U n l i k e some other authors (e.g.,<br />
Weinberg 1972), we will introduce geometrical objects in a coordinate-free manner, only<br />
later introducing coordinates <strong>for</strong> the purpose of simplifying calculations. This approach<br />
requires that we distinguish vec<strong>to</strong>rs from the related objects called one-<strong>for</strong>ms. Once<br />
the dierences and similarities between vec<strong>to</strong>rs, one-<strong>for</strong>ms and tensors are clear, we will<br />
adopt a unied notation that makes computations easy.<br />
2.1 Vec<strong>to</strong>rs<br />
We begin with vec<strong>to</strong>rs. A vec<strong>to</strong>r is a quantity with a magnitude and a direction. This<br />
primitive concept, familiar from undergraduate physics and mathematics, applies equally<br />
in general relativity. An example of a vec<strong>to</strong>r is d~x, the dierence vec<strong>to</strong>r between two<br />
innitesimally close points of spacetime. Vec<strong>to</strong>rs <strong>for</strong>m a linear algebra (i.e., a vec<strong>to</strong>r<br />
~ ~<br />
space). If A is a vec<strong>to</strong>r and a is a real number (scalar) then aA is a vec<strong>to</strong>r with the<br />
same direction (or the opposite direction, if a < 0) whose length is multiplied by jaj. If<br />
~ ~ ~ ~<br />
A and B are vec<strong>to</strong>rs then so is A + B. These results are as valid <strong>for</strong> vec<strong>to</strong>rs in a curved<br />
four-dimensional spacetime as they are <strong>for</strong> vec<strong>to</strong>rs in three-dimensional Euclidean space.<br />
Note that we h a ve i n troduced vec<strong>to</strong>rs without mentioning coordinates or coordinate<br />
trans<strong>for</strong>mations. Scalars and vec<strong>to</strong>rs are invariant under coordinate trans<strong>for</strong>mations<br />
~<br />
vec<strong>to</strong>r components are not. The whole point of writing the laws of physics (e.g., F = m~a)<br />
using scalars and vec<strong>to</strong>rs is that these laws do not depend on the coordinate system<br />
imposed by the physicist.<br />
We denote a spacetime point using a boldface symbo l : x. (This notation is not meant<br />
<strong>to</strong> imply coordinates.) Note that x refers <strong>to</strong> a point, not a vec<strong>to</strong>r. In a curved spacetime<br />
the concept of a radius vec<strong>to</strong>r ~x pointing from some origin <strong>to</strong> each p o i n t x is not useful<br />
because vec<strong>to</strong>rs dened at two dierent points cannot be added straight<strong>for</strong>wardly as<br />
they can in Euclidean space. For example, consider a sphere embedded in ordinary<br />
three-dimensional Euclidean space (i.e., a two-sphere). A v ec<strong>to</strong>r pointing east at one<br />
point on the equa<strong>to</strong>r is seen <strong>to</strong> point radially outward at another point on the equa<strong>to</strong>r<br />
whose longitude is greater by 9 0 . The radially outward direction is undened on the<br />
sphere.<br />
Technically, w e are discussing tangent vec<strong>to</strong>rs that lie in the tangent space of the<br />
manifold at each point. For example, a sphere may b e e m bedded in a three-dimensional<br />
Euclidean space in<strong>to</strong> which m a y be placed a plane tangent <strong>to</strong> the sphere at a point. A two-
dimensional vec<strong>to</strong>r space exists at the point of tangency. H o wever, such a n e m bedding<br />
is not required <strong>to</strong> dene the tangent space of a manifold (Wald 1984). As long as the<br />
space is smooth (as assumed in the <strong>for</strong>mal denition of a manifold), the dierence vec<strong>to</strong>r<br />
d~x be t ween two innitesimally close points may be dened. The set of all d~x denes<br />
the tangent space at x. By assigning a tangent v ec<strong>to</strong>r <strong>to</strong> every spacetime point, we<br />
can recover the usual concept of a vec<strong>to</strong>r eld. However, without additional preparation<br />
one cannot compare vec<strong>to</strong>rs at dierent spacetime points, because they lie in dierent<br />
tangent spaces. In later notes we i n troduce will parallel transport as a means of making<br />
this comparison. Until then, we consider only tangent v ec<strong>to</strong>rs at x. T o emphasize the<br />
status of a tangent v ec<strong>to</strong>r, we will occasionally use a subscript notation: A~<br />
X .<br />
2.2 One-<strong>for</strong>ms and dual vec<strong>to</strong>r space<br />
Next we i n troduce one-<strong>for</strong>ms. A one-<strong>for</strong>m is dened as a linear scalar function of a vec<strong>to</strong>r.<br />
That is, a one-<strong>for</strong>m takes a vec<strong>to</strong>r as input and outputs a scalar. For the one-<strong>for</strong>m P~,<br />
P~(V~ ) is also called the scalar product and may be denoted using angle brackets:<br />
~<br />
P ~ (V ~ ) = hP ~ V i : (1)<br />
The one-<strong>for</strong>m is a linear function, meaning that <strong>for</strong> all scalars a and b and vec<strong>to</strong>rs V~ and<br />
~W , the one-<strong>for</strong>m P ~ satises the following relations:<br />
P ~ (aV ~ + b W ~ ) = hP ~ aV ~ + b Wi ~ = ahP ~ V~ i + bhP ~ W~ i = aP ~ (V ~ ) + bP ~ ( W ~ ) : (2)<br />
Just as we m a y consider any function f( ) as a mathematical entity independently of<br />
any particular argument, we m a y consider the one-<strong>for</strong>m P ~ independently of any particular<br />
vec<strong>to</strong>r V ~ . W e m a y also associate a one-<strong>for</strong>m with each spacetime point, resulting in a<br />
one-<strong>for</strong>m eld P~ = PX ~ . N o w the distinction between a point a v ec<strong>to</strong>r is crucial: P~ X is<br />
a one-<strong>for</strong>m at point x while P~(V~ ) is a scalar, dened implicitly at point x. The scalar<br />
product notation with subscripts makes this more clear: hP ~ X VX ~ i.<br />
One-<strong>for</strong>ms obey their own linear algebra distinct from that of vec<strong>to</strong>rs. Given any t wo<br />
scalars a and b and one-<strong>for</strong>ms P~ and Q, ~ w e m a y dene the one-<strong>for</strong>m aP ~ + b Q ~ by<br />
~ Q)(V~ ) = haP~ + b ~ ~ Q ~ ~ ~ b ~ ~<br />
(aP + b ~ Q V~ i = ahP~ V i + bh ~ V i = aP (V ) + Q(V ) : (3)<br />
Comparing equations (2) and (3), we see that vec<strong>to</strong>rs and one-<strong>for</strong>ms are linear opera<strong>to</strong>rs<br />
on each other, producing scalars. It is often helpful <strong>to</strong> consider a vec<strong>to</strong>r as being a linear<br />
scalar function of a one-<strong>for</strong>m. Thus, we m a y w r i t e hP ~ V~ i = P ~ (V ~ ) = V ~ (P ~ ). The set of<br />
all one-<strong>for</strong>ms is a vec<strong>to</strong>r space distinct from, but complementary <strong>to</strong>, the linear vec<strong>to</strong>r<br />
space of vec<strong>to</strong>rs. The vec<strong>to</strong>r space of one-<strong>for</strong>ms is called the dual vec<strong>to</strong>r (or cotangent)<br />
space <strong>to</strong> distinguish it from the linear space of vec<strong>to</strong>rs (tangent space).<br />
4
5<br />
Although one-<strong>for</strong>ms may appear <strong>to</strong> be highly abstract, the concept of dual vec<strong>to</strong>r<br />
spaces is familiar <strong>to</strong> any student o f q u a n tum mechanics who has seen the Dirac bra-ket<br />
notation. Recall that the fundamental object in quantum mechanics is the state vec<strong>to</strong>r,<br />
represented by a k et j i in a linear vec<strong>to</strong>r space (Hilbert space). A distinct Hilbert<br />
space is given by the set of bra vec<strong>to</strong>rs hj. Bra vec<strong>to</strong>rs and ket vec<strong>to</strong>rs are linear scalar<br />
functions of each other. The scalar product hj i maps a bra vec<strong>to</strong>r and a ket vec<strong>to</strong>r <strong>to</strong> a<br />
scalar called a probability amplitude. The distinction between bras and kets is necessary<br />
because probability amplitudes are complex numbers. As we will see, the distinction<br />
be t ween vec<strong>to</strong>rs and one-<strong>for</strong>ms is necessary because spacetime is curved.<br />
3 <strong>Tensor</strong>s<br />
Having dened vec<strong>to</strong>rs and one-<strong>for</strong>ms we can now dene tensors. A tensor of rank (m n),<br />
also called a (m n) tensor, is dened <strong>to</strong> be a scalar function of m one-<strong>for</strong>ms and n vec<strong>to</strong>rs<br />
that is linear in all of its arguments. It follows at once that scalars are tensors of rank<br />
(0 0), vec<strong>to</strong>rs are tensors of rank (1 0) and one-<strong>for</strong>ms are tensors of rank (0 1). We<br />
may denote a tensor of rank (2 0) by T(P ~ Q) ~ one of rank (2 1) by T(P ~ Q ~ A), ~ etc.<br />
Our notation will not distinguish a (2 0) tensor T from a (2 1) tensor T, although a<br />
notational distinction could be made by placing m arrows and n tildes over the symbo l ,<br />
or by appropriate use of dummy indices (Wald 1984).<br />
The scalar product is a tensor of rank (1 1), which w e will denote I and call the<br />
identity tensor:<br />
I(P ~ V ~ ) h P ~ V ~ i = P ~ (V ~ ) = V ~ (P~) : (4)<br />
We call ~<br />
I the identity because, when applied <strong>to</strong> a xed one-<strong>for</strong>m P ~ and any vec<strong>to</strong>r V , it<br />
returns P~(V~ ). Although the identity t e n s o r w as dened as a mapping from a one-<strong>for</strong>m<br />
and a vec<strong>to</strong>r <strong>to</strong> a scalar, we see that it may equally be interpreted as a mapping from a<br />
one-<strong>for</strong>m <strong>to</strong> the same one-<strong>for</strong>m: I(P ~ ) = P ~ , where the dot indicates that an argument<br />
(a vec<strong>to</strong>r) is needed <strong>to</strong> give a scalar. A similar argument shows that I may be considered<br />
the identity opera<strong>to</strong>r on the space of vec<strong>to</strong>rs V ~ : I( ~V ) = V~ .<br />
A tensor of rank (m n) is linear in all its arguments. For example, <strong>for</strong> (m = 2 n = 0)<br />
we h a ve a straight<strong>for</strong>ward extension of equation (2):<br />
T(aP ~ + b Q ~ c R ~ + dS ~ ) = ac T(P ~ R) ~ + ad T(P ~ S) ~ + bc T( Q ~ R) ~ + bd T( q ~ S ~ ) : (5)<br />
<strong>Tensor</strong>s of a given rank <strong>for</strong>m a linear algebra, meaning that a linear combination of<br />
tensors of rank (m n) is also a tensor of rank (m n), dened by straight<strong>for</strong>ward extension<br />
of equation (3). Two tensors (of the same rank) are equal if and only if they return the<br />
same scalar when applied <strong>to</strong> all possible input vec<strong>to</strong>rs and one-<strong>for</strong>ms. <strong>Tensor</strong>s of dierent<br />
rank cannot be added or compared, so it is important t o k eep track of the rank of each
tensor. Just as in the case of scalars, vec<strong>to</strong>rs and one-<strong>for</strong>ms, tensor elds TX are dened<br />
by associating a tensor with each spacetime point.<br />
There are three ways <strong>to</strong> change the rank of a tensor. The rst, called the tensor (or<br />
outer) product, combines two tensors of ranks (m 1 n 1) and ( m 2 n 2) <strong>to</strong> <strong>for</strong>m a tensor<br />
of rank (m 1 + m 2 n 1 + n 2 ) b y simply combining the argument lists of the two tensors<br />
and thereby expanding the dimensionality of the tensor space. For example, the tensor<br />
product of two v ec<strong>to</strong>rs A ~ and B ~ gives a rank (2 0) tensor<br />
T = A ~ B~ T(P ~ Q) ~ A(P~) ~ B( ~ Q) ~ : (6)<br />
We use the symbol <strong>to</strong> denote the tensor product later we will drop this symbo l f o r<br />
notational convenience when it is clear from the context that a tensor product is implied.<br />
Note that the tensor product is non-commutative: A ~ B ~ = 6 B ~ A ~ (unless B ~ = cA ~ <strong>for</strong><br />
some scalar c) because A(P~) ~ B( ~ Q) ~ = 6 A( ~ Q) ~ B(P~) ~ <strong>for</strong> all P ~ and Q. ~ W e use the symbo l <br />
<strong>to</strong> denote the tensor product of any t wo tensors, e.g., P ~ ~ ~ ~<br />
T = P A B is a tensor<br />
of rank (2 1). The second way t o c hange the rank of a tensor is by c o n traction, which<br />
reduces the rank of a (m n) tensor <strong>to</strong> (m ; 1 n ; 1). The third way is the gradient. We<br />
will discuss contraction and gradients later.<br />
3.1 Metric tensor<br />
The scalar product (eq. 1) requires a vec<strong>to</strong>r and a one-<strong>for</strong>m. Is it possible <strong>to</strong> obtain a<br />
scalar from two v ec<strong>to</strong>rs or two one-<strong>for</strong>ms From the denition of tensors, the answer is<br />
clearly yes. Any tensor of rank (0 2) will give a scalar from two v ec<strong>to</strong>rs and any tensor<br />
of rank (2 0) combines two one-<strong>for</strong>ms <strong>to</strong> give a scalar. However, there is a particular<br />
;1<br />
(0 2) tensor eld gX called the metric tensor and a related (2 0) tensor eld g<br />
X called<br />
the inverse metric tensor <strong>for</strong> which special distinction is reserved. The metric tensor is<br />
a symmetric bilinear scalar function of two v ec<strong>to</strong>rs. That is, given vec<strong>to</strong>rs V~ and W ~ , g<br />
returns a scalar, called the dot product:<br />
~ ~ ~ ~ ~<br />
g(V ~ W ) = V W = W ~ V = g(W ~ V ) : (7)<br />
;1<br />
Similarly, g returns a scalar from two one-<strong>for</strong>ms P ~ and Q, ~ w h ich w e also call the dot<br />
product:<br />
;1<br />
g (P ~ Q) ~ = ~ ~ ~ ~ ;1<br />
P Q = P Q = g (P ~ Q) ~ : (8)<br />
Although a dot is used in both cases, it should be clear from the context whether g or g<br />
;1<br />
is implied. We reserve the dot product notation <strong>for</strong> the metric and inverse metric tensors<br />
just as we reserve the angle brackets scalar product notation <strong>for</strong> the identity tensor (eq.<br />
4). Later (in eq. 41) we will see what distinguishes g from other (0 2) tensors and g<br />
;1<br />
from other symmetric (2 0) tensors.<br />
6
One of the most important properties of the metric is that it allows us <strong>to</strong> convert<br />
~<br />
vec<strong>to</strong>rs <strong>to</strong> one-<strong>for</strong>ms. If we <strong>for</strong>get <strong>to</strong> include W in equation (7) we get a quantity, denoted<br />
V ~ , that behaves like a one-<strong>for</strong>m:<br />
~V ( ) g(V ~ ) = g( V ~ ) (9)<br />
where we h a ve inserted a dot <strong>to</strong> remind ourselves that a vec<strong>to</strong>r must be inserted <strong>to</strong> give<br />
a scalar. (Recall that a one-<strong>for</strong>m is a scalar function of a vec<strong>to</strong>r!) We use the same letter<br />
<strong>to</strong> indicate the relation of V ~ and V ~ .<br />
Thus, the metric g is a mapping from the space of vec<strong>to</strong>rs <strong>to</strong> the space of one-<strong>for</strong>ms:<br />
~ ~ ;1 ;1<br />
g : V ! V . By denition, the inverse metric g is the inverse mapping: g : V ~ ! V ~ .<br />
(The inverse always exists <strong>for</strong> nonsingular spacetimes.) Thus, if V~ is dened <strong>for</strong> any V ~<br />
by equation (9), the inverse metric tensor is dened by<br />
~ ~<br />
V ( ) g<br />
;1<br />
(V ) = g<br />
;1<br />
( V ~ ) : (10)<br />
Equations (4) and (7){(10) give us several equivalent w ays <strong>to</strong> obtain scalars from vec<strong>to</strong>rs<br />
~ ~ ~<br />
V and W and their associated one-<strong>for</strong>ms V ~ and W :<br />
hV ~ Wi ~ = hW ~ V~ i = V~ W ~ = V~ W ~ = I(V ~ W) ~ = I(W ~ V ~ ) = g(V ~ ~ ;1<br />
W ) = g (V ~ W ~ ) : (11)<br />
3.2 Basis vec<strong>to</strong>rs and one-<strong>for</strong>ms<br />
It is possible <strong>to</strong> <strong>for</strong>mulate the mathematics of general relativity e n tirely using the abstract<br />
<strong>for</strong>malism of vec<strong>to</strong>rs, <strong>for</strong>ms and tensors. However, while the geometrical (coordinate-free)<br />
interpretation of quantities should always be kept in mind, the abstract approach often is<br />
not the most practical way <strong>to</strong> per<strong>for</strong>m calculations. To simplify calculations it is helpful<br />
<strong>to</strong> introduce a set of linearly independent basis vec<strong>to</strong>r and one-<strong>for</strong>m elds spanning<br />
our vec<strong>to</strong>r and dual vec<strong>to</strong>r spaces. In the same way, practical calculations in quantum<br />
mechanics often start by expanding the ket vec<strong>to</strong>r in a set of basis kets, e.g., energy<br />
eigenstates. By denition, the dimensionality of spacetime (four) equals the numbe r o f<br />
linearly independent basis vec<strong>to</strong>rs and one-<strong>for</strong>ms.<br />
We denote our set of basis vec<strong>to</strong>r elds by f~e X g, where labels the basis vec<strong>to</strong>r<br />
(e.g., = 0 1 2 3) and x labels the spacetime point. Any four linearly independent basis<br />
vec<strong>to</strong>rs at each spacetime point will work we do not not impose orthonormality o r a n y<br />
other conditions in general, nor have w e implied any relation <strong>to</strong> coordinates (although<br />
later we will). Given a basis, we m a y expand any v ec<strong>to</strong>r eld A~<br />
as a linear combination<br />
of basis vec<strong>to</strong>rs:<br />
~ <br />
AX = A X ~e X = A 0 X~e 0 X + A 1 X~e 1 X + A 2 X~e 2 X + A 3 X~e 3 X : (12)<br />
7
Note our placement of subscripts and superscripts, chosen <strong>for</strong> consistency with the Einstein<br />
summation convention, which requires pairing one subscript with one superscript.<br />
The coecients A are called the components of the vec<strong>to</strong>r (often, the contravariant<br />
~<br />
components). Note well that the coecients A depend on the basis vec<strong>to</strong>rs but A does<br />
not!<br />
Similarly, w e m a y c hoose a basis of one-<strong>for</strong>m elds in which <strong>to</strong> expand one-<strong>for</strong>ms<br />
like A~ X . Although any set of four linearly independent one-<strong>for</strong>ms will suce <strong>for</strong> each<br />
spacetime point, we prefer <strong>to</strong> choose a special one-<strong>for</strong>m basis called the dual basis and<br />
<br />
denoted fe~ X g. Note that the placement of subscripts and superscripts is signicant<br />
we never use a subscript <strong>to</strong> label a basis one-<strong>for</strong>m while we n e v er use a superscript<br />
<strong>to</strong> label a basis vec<strong>to</strong>r. There<strong>for</strong>e, e~ is not related <strong>to</strong> ~e through the metric (eq. 9):<br />
e~ ( ) 6= g(~e ). Rather, the dual basis one-<strong>for</strong>ms are dened by imposing the following<br />
16 requirements at each spacetime point:<br />
he~ X ~e Xi = <br />
<br />
(13)<br />
where <br />
is the Kronecker delta, <br />
= 1 if = and <br />
= 0 otherwise, with the<br />
same values <strong>for</strong> each spacetime point. (We m ust always distinguish subscripts from<br />
superscripts the Kronecker delta always has one of each.) Equation (13) is a system of<br />
four linear equations at each spacetime point <strong>for</strong> each of the four quantities e~ and it<br />
has a unique solution. (The reader may s h o w t h a t a n y n o n trivial trans<strong>for</strong>mation of the<br />
dual basis one-<strong>for</strong>ms will violate eq. 13.) Now we ma y expand any one-<strong>for</strong>m eld P~ X in<br />
the basis of one-<strong>for</strong>ms:<br />
P ~ X = P X e~ X : (14)<br />
The component P of the one-<strong>for</strong>m P ~ is often called the covariant component <strong>to</strong> distin-<br />
~<br />
guish it from the contravariant component P<br />
<br />
of the vec<strong>to</strong>r P . In fact, because we h a ve<br />
consistently treated vec<strong>to</strong>rs and one-<strong>for</strong>ms as distinct, we should not think of these as<br />
being distinct "components" of the same entity a t a l l .<br />
There is a simple way <strong>to</strong> get the components of vec<strong>to</strong>rs and one-<strong>for</strong>ms, using the fact<br />
that vec<strong>to</strong>rs are scalar functions of one-<strong>for</strong>ms and vice versa. One simply evaluates the<br />
vec<strong>to</strong>r using the appropriate basis one-<strong>for</strong>m:<br />
~ <br />
~ <br />
A( e~ ) = he~ Ai = he~ A ~e i = he~ ~e iA <br />
= A <br />
= A <br />
(15)<br />
and conversely <strong>for</strong> a one-<strong>for</strong>m:<br />
P ~ (~e ) = hP<br />
~ <br />
<br />
~e i = hP e~ ~e i = he~ ~e iP = P = P : (16)<br />
We h a ve suppressed the spacetime point x <strong>for</strong> clarity, but it is always implied.<br />
8
3.3 <strong>Tensor</strong> algebra<br />
We can use the same ideas <strong>to</strong> expand tensors as products of components and basis<br />
tensors. First we note that a basis <strong>for</strong> a tensor of rank (m n) i s p r o vided by the tensor<br />
product of m vec<strong>to</strong>rs and n one-<strong>for</strong>ms. For example, a (0 2) tensor like the metric tensor<br />
<br />
can be decomposed in<strong>to</strong> basis tensors e~ e~ . T he components of a tensor of rank (m n),<br />
labeled with m superscripts and n subscripts, are obtained by e v aluating the tensor using<br />
m basis one-<strong>for</strong>ms and n basis vec<strong>to</strong>rs. For example, the components of the (0 2) metric<br />
tensor, the (2 0) inverse metric tensor and the (1 1) identity tensor are<br />
;1 <br />
g g(~e ~e ) = ~e ~e g g ( e~ e~ ) = e~ e~ <br />
<br />
= I( e~ ~e ) = he~ ~e i : (17)<br />
(The last equation follows from eqs. 4 and 13.) The tensors are given by summing over<br />
the tensor product of basis vec<strong>to</strong>rs and one-<strong>for</strong>ms:<br />
;1<br />
g = g e~ <br />
e~ g = g ~e ~e I = <br />
~e e~ <br />
: (18)<br />
The reader should check that equation (18) follows from equations (17) and the duality<br />
condition equation (13).<br />
Basis vec<strong>to</strong>rs and one-<strong>for</strong>ms allow us <strong>to</strong> represent a n y tensor equations using components.<br />
For example, the dot product between two v ec<strong>to</strong>rs or two one-<strong>for</strong>ms and the<br />
scalar product between a one-<strong>for</strong>m and a vec<strong>to</strong>r may be written using components as<br />
~ ~ ~ ~ <br />
A B = g A A hP~ Ai = P A P~ Q = g P P : (19)<br />
The reader should prove these important results.<br />
If two tensors of the same rank are equal in one basis, i.e., if all of their components<br />
are equal, then they are equal in any basis. While this mathematical result is obvious<br />
from the basis-free meaning of a tensor, it will have important p h ysical implications in<br />
GR arising from the Equivalence Principle.<br />
As we discussed above, the metric and inverse metric tensors allow us <strong>to</strong> trans<strong>for</strong>m<br />
vec<strong>to</strong>rs in<strong>to</strong> one-<strong>for</strong>ms and vice versa. If we e v aluate the components of V ~ and the<br />
one-<strong>for</strong>m V~ dened by equations (9) and (10), we g e t<br />
V = g(~e V ~ ) = g V <br />
;1 ~ <br />
V = g ( e~ V ) = g : (20)<br />
V <br />
Because these two equations must hold <strong>for</strong> any v ec<strong>to</strong>r V~ , w e conclude that the matrix<br />
dened by g is the inverse of the matrix dened by g :<br />
<br />
g g = : (21)<br />
We also see that the metric and its inverse are used <strong>to</strong> lower and raise indices on compo-<br />
~ ~<br />
nents. Thus, given two v ec<strong>to</strong>rs V and W , w e m a y e v aluate the dot product any of four<br />
equivalent w ays (cf. eq. 11):<br />
~ ~ <br />
V W = g V W = V W = V W = g V W : (22)<br />
9
10<br />
In fact, the metric and its inverse may be used <strong>to</strong> trans<strong>for</strong>m tensors of rank (m n)<br />
in<strong>to</strong> tensors of any rank ( m + k n ; k) where k = ;m ;m + 1 : : : n . Consider, <strong>for</strong><br />
example, a (1 2) tensor T with components<br />
<br />
T T( e~ ~e ~e ) : (23)<br />
If we fail <strong>to</strong> plug in the one-<strong>for</strong>m ~e <br />
, the result is the vec<strong>to</strong>r T ~e . (A one-<strong>for</strong>m must be<br />
<br />
inserted <strong>to</strong> return the quantity T .) This vec<strong>to</strong>r may then be inserted in<strong>to</strong> the metric<br />
tensor <strong>to</strong> give the components of a (0 3) tensor:<br />
T g(~e T<br />
<br />
<br />
~e ) = g T<br />
<br />
<br />
: (24)<br />
We could now use the inverse metric <strong>to</strong> raise the third index, say, giving us the component<br />
of a (1 2) tensor distinct from equation (23):<br />
T <br />
<br />
g<br />
;1<br />
( e~ T e~ ) = g<br />
T = g g T<br />
<br />
<br />
<br />
<br />
: (25)<br />
In fact, there are 2 m+n dierent tensor spaces with ranks summing <strong>to</strong> m + n. The metric<br />
or inverse metric tensor allow all of these tensors <strong>to</strong> be trans<strong>for</strong>med in<strong>to</strong> each other.<br />
Returning <strong>to</strong> equation (22), we see w h y we m ust distinguish vec<strong>to</strong>rs (with components<br />
<br />
V ) from one-<strong>for</strong>ms (with components V ). The scalar product of two v ec<strong>to</strong>rs requires<br />
the metric tensor while that of two one-<strong>for</strong>ms requires the inverse metric tensor. In<br />
general, g 6= g . The only case in which the distinction is unnecessary is in at<br />
(Lorentz) spacetime with orthonormal Cartesian basis vec<strong>to</strong>rs, in which ca se g = <br />
is everywhere the diagonal matrix with entries (;1 +1 +1 +1). However, gravity curves<br />
spacetime. (Besides, we m a y wish <strong>to</strong> use curvilinear coordinates even in at spacetime.)<br />
As a result, it is impossible <strong>to</strong> dene a coordinate system <strong>for</strong> which g = g everywhere.<br />
We m ust there<strong>for</strong>e distinguish vec<strong>to</strong>rs from one-<strong>for</strong>ms and we m ust be careful about the<br />
placement of subscripts and superscripts on components.<br />
At this stage it is useful <strong>to</strong> introduce a classication of vec<strong>to</strong>rs and one-<strong>for</strong>ms drawn<br />
from special relativity with its Minkowski metric . Recall that a vec<strong>to</strong>r A ~ = A ~e <br />
is called spacelike, timelike o r n ull according <strong>to</strong> whether A ~ A ~ = A A is positive,<br />
negative or zero, respectively. In a Euclidean space, with positive denite metric, A ~ A ~<br />
is never negative. However, in the Lorentzian spacetime geometry of special relativity,<br />
time enters the metric with opposite sign so that it is possible <strong>to</strong> have A ~ A ~ < 0. In<br />
particular, the four-velocity u = dx =d of a massive particle (where d is proper time)<br />
is a timelike v ec<strong>to</strong>r. This is seen most simply by per<strong>for</strong>ming a Lorentz boost <strong>to</strong> the<br />
z<br />
<br />
rest frame of the particle in which c a s e u t = 1, u x = u y = u = 0 and u u = ;1.<br />
<br />
Now, u u is a Lorentz scalar so that u ~ ~ u = ;1 i n a n y Lorentz frame. Often this is<br />
written p~ p~ = ;m 2 where p = mu is the four-momentum <strong>for</strong> a particle of mass m.<br />
For a massless particle (e.g., a pho<strong>to</strong>n) the proper time vanishes but the four-momentum
11<br />
is still well-dened with p~ p~ = 0: the momentum vec<strong>to</strong>r is null. We adopt the same<br />
notation in general relativity, replacing the Minkowski metric (components ) with the<br />
actual metric g and evaluating the dot product using A ~ A ~ = g(A ~ A) ~ = g A A . The<br />
;1<br />
same classication scheme extends <strong>to</strong> one-<strong>for</strong>ms using g : a one-<strong>for</strong>m P ~ is spacelike,<br />
;1<br />
timelike o r n ull according <strong>to</strong> whether P~ P~ = g (P ~ P ~) = g P P is positive, negative<br />
or zero, respectively. Finally, a v ec<strong>to</strong>r is called a unit vec<strong>to</strong>r if A ~ A ~ = 1 and similarly<br />
<strong>for</strong> a one-<strong>for</strong>m. The four-velocity of a massive particle is a timelike u n i t v ec<strong>to</strong>r.<br />
Now that we h a ve i n troduced basis vec<strong>to</strong>rs and one-<strong>for</strong>ms, we can dene the contraction<br />
of a tensor. Contraction pairs two argument s o f a r a n k ( m n) tensor: one vec<strong>to</strong>r and<br />
one one-<strong>for</strong>m. The arguments are replaced by b a s i s v ec<strong>to</strong>rs and one-<strong>for</strong>ms and summed<br />
over. For example, consider the (1 3) tensor R, which m a y b e c o n tracted on its second<br />
vec<strong>to</strong>r argument <strong>to</strong> give a ( 0 2) tensor also denoted R but distinguished by its shorter<br />
argument list:<br />
X<br />
3<br />
R(A ~ B) ~ = ~ B) = R( e~ A~e ~ ~<br />
R( e~ A ~e ~ B) : (26)<br />
In later notes we will dene the Riemann curvature tensor of rank (1 3) its contraction,<br />
dened by equation (26), is called the Ricci tensor. Although the contracted tensor would<br />
appear <strong>to</strong> depend on the choice of basis because its denition involves the basis vec<strong>to</strong>rs<br />
and one-<strong>for</strong>ms, the reader may s h o w that it is actually invariant under a change of basis<br />
(and is there<strong>for</strong>e a tensor) as long as we use dual one-<strong>for</strong>m and vec<strong>to</strong>r bases satisfying<br />
equation (13). Equation (26) becomes somewhat clearer if we e x p r e s s i t e n tirely using<br />
tensor components:<br />
R = R : (27)<br />
Summation over is implied. Contraction may be per<strong>for</strong>med on any pair of covariant<br />
and contravariant indices dierent tensors result.<br />
=0